8 The five letters of the word NEVER are arranged in random order in a straight line.
- How many different orders of the letters are possible?
- In how many of the possible orders are the two Es next to each other?
- Find the probability that the first two letters in the order include exactly one letter E.
\(9 R\) and \(S\) are independent random variables each having the distribution \(\operatorname { Geo } ( p )\). - Find \(\mathrm { P } ( R = 1\) and \(S = 1 )\) in terms of \(p\).
- Show that \(\mathrm { P } ( R = 3\) and \(S = 3 ) = p ^ { 2 } q ^ { 4 }\), where \(q = 1 - p\).
- Use the formula for the sum to infinity of a geometric series to show that
$$\mathrm { P } ( R = S ) = \frac { p } { 2 - p }$$